arXiv subjects

Argyrios Deligkas

Publications and source records attributed to Argyrios Deligkas.

At least 55 records · Page 3Linked to original sources

arXiv2022

The Parameterized Complexity of Welfare Guarantees in Schelling Segregation

Schelling's model considers $k$ types of agents each of whom needs to select a vertex on an undirected graph, where every agent prefers to neighbor agents of the same type. We are motivated by a recent line of work that studies solutions that are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where $k=2$ and there are $r$ red agents and $b$ blue agents. We show that all solution-notions we consider are $\textsf{NP}$-hard to compute even when $b=1$ and that they are $\textsf{W}[1]$-hard when parameterized by $r$ and $b$. In addition, we show that WO and GWO are $\textsf{NP}$-hard even on cubic graphs. We complement these negative results by an $\textsf{FPT}$ algorithm parameterized by $r, b$ and the maximum degree of the graph. For the general case with $k$ types of agents, we prove that for any of the notions we consider the problem is $\textsf{W}[1]$-hard when parameterized by $k$ for a large family of graphs that includes trees. We accompany these negative results with an $\textsf{XP}$ algorithm parameterized by $k$ and the treewidth of the graph.

cs.DS↗

arXiv2021

Combinatorial Algorithms for Multidimensional Necklaces

A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting, generating, ranking, and unranking. This paper generalises the concept of necklaces to the multidimensional setting. We define multidimensional necklaces as an equivalence classes over multidimensional words under the multidimensional cyclic shift operation. Alongside this definition, we generalise several problems from the one dimensional setting to the multidimensional setting for multidimensional necklaces with size $(n_1,n_2,...,n_d)$ over an alphabet of size $q$ including: providing closed form equations for counting the number of necklaces; an $O(n_1 \cdot n_2 \cdot ... \cdot n_d)$ time algorithm for transforming some necklace $w$ to the next necklace in the ordering; an $O((n_1 \cdot n_2 \cdot ... \cdot n_d)^5)$ time algorithm to rank necklaces (determine the number of necklaces smaller than $w$ in the set of necklaces); an $O((n_1\cdot n_2 \cdot ... \cdot n_d)^{6(d + 1)} \cdot \log^d(q))$ time algorithm to unrank multidimensional necklace (determine the $i^{th}$ necklace in the set of necklaces). Our results on counting, ranking, and unranking are further extended to the fixed content setting, where every necklace has the same Parikh vector, in other words every necklace shares the same number of occurrences of each symbol. Finally, we study the $k$-centre problem for necklaces both in the single and multidimensional settings. We provide strong approximation algorithms for solving this problem in both the one dimensional and multidimensional settings.

math.CO↗

arXiv2021

Heterogeneous Facility Location with Limited Resources

We initiate the study of the heterogeneous facility location problem with limited resources. We mainly focus on the fundamental case where a set of agents are positioned in the line segment [0,1] and have approval preferences over two available facilities. A mechanism takes as input the positions and the preferences of the agents, and chooses to locate a single facility based on this information. We study mechanisms that aim to maximize the social welfare (the total utility the agents derive from facilities they approve), under the constraint of incentivizing the agents to truthfully report their positions and preferences. We consider three different settings depending on the level of agent-related information that is public or private. For each setting, we design deterministic and randomized strategyproof mechanisms that achieve a good approximation of the optimal social welfare, and complement these with nearly-tight impossibility results.

cs.GT↗

arXiv2021

Walrasian Equilibria in Markets with Small Demands

We study the complexity of finding a Walrasian equilibrium in markets where the agents have $k$-demand valuations. These valuations are an extension of unit-demand valuations where a bundle's value is the maximum of its $k$-subsets' values. For unit-demand agents, where the existence of a Walrasian equilibrium is guaranteed, we show that the problem is in quasi-NC. For $k=2$, we show that it is NP-hard to decide if a Walrasian equilibrium exists even if the valuations are fractionally subadditive (XOS), while for $k=3$ the hardness carries over to budget-additive valuations. In addition, we give a polynomial-time algorithm for markets with 2-demand single-minded valuations, or unit-demand valuations.

cs.GT↗

arXiv2021

Ranking Bracelets in Polynomial Time

The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm.

math.CO↗

arXiv2021

Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem

We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete, we show that the exact version is much harder. Specifically, finding a solution with $n$ cuts is FIXP-hard, and deciding whether there exists a solution with fewer than $n$ cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP $\subseteq$ BU $\subseteq$ TFETR and that LinearBU $=$ PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.

cs.CC↗

arXiv2020

Optimizing Reachability Sets in Temporal Graphs by Delaying

A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. The questions about reachability sets are motivated by numerous applications of temporal graphs in network epidemiology, which aim to minimise the spread of infection, and scheduling problems in supply networks in manufacturing with the opposite objectives of maximising coverage and productivity. We introduce control mechanisms for reachability sets that are based on two natural operations of delaying. The first operation, termed merging, is global and batches together consecutive time labels into a single time label in the whole network simultaneously. This corresponds to postponing all events until a particular time. The second, imposes independent delays on the time labels of every edge of the graph. We provide a thorough investigation of the computational complexity of different objectives related to reachability sets when these operations are used. For the merging operation, i.e. global lockdown effect, we prove NP-hardness results for several minimization and maximization reachability objectives, even for very simple graph structures. For the second operation, independent delays, we prove that the minimization problems are NP-hard when the number of allowed delays is bounded. We complement this with a polynomial-time algorithm for minimising the reachability set in case of unbounded delays.

cs.DS↗

arXiv2020

Heterogeneous Facility Location Games

We study heterogeneous $k$-facility location games. In this model there are $k$ facilities where each facility serves a different purpose. Thus, the preferences of the agents over the facilities can vary arbitrarily. Our goal is to design strategy proof mechanisms that place the facilities in a way to maximize the minimum utility among the agents. For $k=1$, if the agents' locations are known, we prove that the mechanism that places the facility on an optimal location is strategy proof. For $k \geq 2$, we prove that there is no optimal strategy proof mechanism, deterministic or randomized, even when $k=2$ there are only two agents with known locations, and the facilities have to be placed on a line segment. We derive inapproximability bounds for deterministic and randomized strategy proof mechanisms. Finally, we focus on the line segment and provide strategy proof mechanisms that achieve constant approximation. All of our mechanisms are simple and communication efficient. As a byproduct we show that some of our mechanisms can be used to achieve constant factor approximations for other objectives as the social welfare and the happiness.

cs.GT↗

arXiv2020

Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle

In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph $G$ and a Hamiltonian cycle $C_0$ of $G$, how can we compute a second Hamiltonian cycle $C_1 \neq C_0$ of $G$? Cedric Smith proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is an algorithm which computes the second Hamiltonian cycle in time $O(n \cdot 2^{(0.3-\varepsilon)n})$ time, for some positive constant $\varepsilon>0$, and in polynomial space, thus improving the state of the art running time for solving this problem. Our algorithm is based on a fundamental structural property of Thomason's lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a Hamiltonian graph $G$ with a given Hamiltonian cycle $C_0$ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least $n - 4α(\sqrt{n}+2α)+8$, where $α= \frac{Δ-2}{δ-2}$ and $δ,Δ$ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.

cs.DS↗

arXiv2020

The K-Centre Problem for Necklaces

In graph theory, the objective of the k-centre problem is to find a set of $k$ vertices for which the largest distance of any vertex to its closest vertex in the $k$-set is minimised. In this paper, we introduce the $k$-centre problem for sets of necklaces, i.e. the equivalence classes of words under the cyclic shift. This can be seen as the k-centre problem on the complete weighted graph where every necklace is represented by a vertex, and each edge has a weight given by the overlap distance between any pair of necklaces. Similar to the graph case, the goal is to choose $k$ necklaces such that the distance from any word in the language and its nearest centre is minimised. However, in a case of k-centre problem for languages the size of associated graph maybe exponential in relation to the description of the language, i.e., the length of the words l and the size of the alphabet q. We derive several approximation algorithms for the $k$-centre problem on necklaces, with logarithmic approximation factor in the context of l and k, and within a constant factor for a more restricted case.

cs.DS↗

arXiv2020

Matching in Stochastically Evolving Graphs

This paper studies the maximum cardinality matching problem in stochastically evolving graphs. We formally define the arrival-departure model with stochastic departures. There, a graph is sampled from a specific probability distribution and it is revealed as a series of snapshots. Our goal is to study algorithms that create a large matching in the sampled graphs. We define the price of stochasticity for this problem which intuitively captures the loss of any algorithm in the worst case in the size of the matching due to the uncertainty of the model. Furthermore, we prove the existence of a deterministic optimal algorithm for the problem. In our second set of results we show that we can efficiently approximate the expected size of a maximum cardinality matching by deriving a fully randomized approximation scheme (FPRAS) for it. The FPRAS is the backbone of a probabilistic algorithm that is optimal when the model is defined over two timesteps. Our last result is an upper bound of $\frac{2}{3}$ on the price of stochasticity. This means that there is no algorithm that can match more than $\frac{2}{3}$ of the edges of an optimal matching in hindsight.

cs.DS↗

arXiv2020

Crystal Structure Prediction via Oblivious Local Search

We study Crystal Structure Prediction, one of the major problems in computational chemistry. This is essentially a continuous optimization problem, where many different, simple and sophisticated, methods have been proposed and applied. The simple searching techniques are easy to understand, usually easy to implement, but they can be slow in practice. On the other hand, the more sophisticated approaches perform well in general, however almost all of them have a large number of parameters that require fine tuning and, in the majority of the cases, chemical expertise is needed in order to properly set them up. In addition, due to the chemical expertise involved in the parameter-tuning, these approaches can be {\em biased} towards previously-known crystal structures. Our contribution is twofold. Firstly, we formalize the Crystal Structure Prediction problem, alongside several other intermediate problems, from a theoretical computer science perspective. Secondly, we propose an oblivious algorithm for Crystal Structure Prediction that is based on local search. Oblivious means that our algorithm requires minimal knowledge about the composition we are trying to compute a crystal structure for. In addition, our algorithm can be used as an intermediate step by {\em any} method. Our experiments show that our algorithms outperform the standard basin hopping, a well studied algorithm for the problem.

cs.CE↗

arXiv2020

Approximating the Existential Theory of the Reals

The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of variables in $\mathbb{R}$. In this paper we propose and study the approximate existential theory of the reals ($ε$-ETR), in which the constraints only need to be satisfied approximately. We first show that when the domain of the variables is $\mathbb{R}$ then $ε$-ETR = ETR under polynomial time reductions, and then study the constrained $ε$-ETR problem when the variables are constrained to lie in a given bounded convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It discretizes the domain in a grid-like manner whose density depends on various properties of the formula. A consequence of our theorem is that we obtain a quasi-polynomial time approximation scheme (QPTAS) for a fragment of constrained $ε$-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.

cs.CC↗

arXiv2019

Connected Subgraph Defense Games

We study a security game over a network played between a $defender$ and $k$ $attackers$. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $λ$ nodes to scan and clean.Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. "The price of defense" (MFCS'06). We are interested in Nash equilibria (NE) of this game, as well as in characterizing $defense$-$optimal$ networks which allow for the best $equilibrium$ $defense$ $ratio$, termed Price of Defense; this is the ratio of $k$ over the expected number of attackers that the defender catches in a NE. We provide characterizations of the NEs of this game and defense-optimal networks. This allows us to show that the NEs of the game coincide independently from the coordination or not of the attackers. In addition, we give an algorithm for computing NEs. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $λ$ constantly close to 1 or $n$. For the special case of tree-networks, we refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal NE. On the other hand, we prove that it is NP-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any $λ$.

cs.GT↗

arXiv2019

Directed Graph Minors and Serial-Parallel Width

Graph minors are a primary tool in understanding the structure of undirected graphs, with many conceptual and algorithmic implications. We propose new variants of \emph{directed graph minors} and \emph{directed graph embeddings}, by modifying familiar definitions. For the class of 2-terminal directed acyclic graphs (TDAGs) our two definitions coincide, and the class is closed under both operations. The usefulness of our directed minor operations is demonstrated by characterizing all TDAGs with serial-parallel width at most $k$; a class of networks known to guarantee bounded negative externality in nonatomic routing games. Our characterization implies that a TDAG has serial-parallel width of $1$ if and only if it is a directed series-parallel graph. We also study the computational complexity of finding a directed minor and computing the serial-parallel width.

cs.DM↗

arXiv2018

Binary Search in Graphs Revisited

In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates' set for the target, while each query asks an appropriately chosen vertex-- the "median"--which minimises a potential $Φ$ among the vertices of the candidates' set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates' set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential $Γ$ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential $Γ$ that allows querying approximate medians.

cs.DS↗

arXiv2017

Computing Constrained Approximate Equilibria in Polymatrix Games

This paper is about computing constrained approximate Nash equilibria in polymatrix games, which are succinctly represented many-player games defined by an interaction graph between the players. In a recent breakthrough, Rubinstein showed that there exists a small constant $ε$, such that it is PPAD-complete to find an (unconstrained) $ε$-Nash equilibrium of a polymatrix game. In the first part of the paper, we show that is NP-hard to decide if a polymatrix game has a constrained approximate equilibrium for 9 natural constraints and any non-trivial approximation guarantee. These results hold even for planar bipartite polymatrix games with degree 3 and at most 7 strategies per player, and all non-trivial approximation guarantees. These results stand in contrast to similar results for bimatrix games, which obviously need a non-constant number of actions, and which rely on stronger complexity-theoretic conjectures such as the exponential time hypothesis. In the second part, we provide a deterministic QPTAS for interaction graphs with bounded treewidth and with logarithmically many actions per player that can compute constrained approximate equilibria for a wide family of constraints that cover many of the constraints dealt with in the first part.

cs.GT↗

arXiv2017

Inapproximability Results for Approximate Nash Equilibria

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem $ε$-NE $δ$-SW: find an $ε$-approximate Nash equilibrium ($ε$-NE) that is within $δ$ of the best social welfare achievable by an $ε$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(δ)\right)$-NE $\mathrm{O}(δ)$-SW for an $n\times n$ bimatrix game requires $n^{\mathrm{\widetilde Ω}(\log n)}$ time. Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player's payoff, or finding approximate equilibria contained in a given pair of supports. We show quasi-polynomial lower bounds for these problems assuming that ETH holds, where these lower bounds apply to $ε$-Nash equilibria for all $ε< \frac{1}{8}$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.

cs.GT↗